Spherical and geodesic growth rates of right-angled Coxeter and Artin groups are Perron numbers

Abstract

We prove that for any infinite right-angled Coxeter or Artin group, its spherical and geodesic growth rates (with respect to the standard generating set) either take values in the set of Perron numbers, or equal $1$. Also, we compute the average number of geodesics representing an element of given word length in such groups.